We present MathSAT 4, a state-of-the-art SMT solver. MathSAT 4 handles several useful theories: (combinations of) equality and uninterpreted functions, difference logic, linear arithmetic, and the theory of bit-vectors. It was explicitly designed for being used in formal verification, and thus provides functionalities which extend the applicability of SMT in this setting. In particular: model generation (for counterexample reconstruction), model enumeration (for predicate abstraction), an incremental interface (for BMC), and computation of unsatisfiable cores and Craig interpolants (for abstraction refinement).

Many approaches to deciding the satisfiability of quantifier-free formulae with respect to a background theory T-also known as Satisfiability Modulo Theory, or SMT(T)-rely on the integration between an enumerator of truth assignments and a decision procedure for conjunction of literals in T. When the background theory T is the combination T 1 ∪ T 2 of two simpler theories, the approach is typically instantiated by means of a theory combination schema (e.g. Nelson-Oppen, Shostak). In this paper we propose a new approach to SMT(T 1 ∪ T 2), where the enumerator of truth assignments is integrated with two decision procedures, one for T 1 and one for T 2 , acting independently from each other. The key idea is to search for a truth assignment not only to the atoms occurring in the formula, but also to all the equalities between variables which are shared between the theories. This approach is simple and expressive: for instance, no modification is required to handle non-convex theories (as opposed to traditional Nelson-Oppen combinations which require a mechanism for splitting). Furthermore, it can be made practical by leveraging on state-of-the-art boolean and SMT

The use of interpolants in verification is gaining more and more importance. Since theories used in applications are usually obtained as (disjoint) combinations of simpler theories, it is important to modularly reuse interpolation algorithms for the component theories. We show that a sufficient and necessary condition to do this for quantifier-free interpolation is that the component theories have the strong (sub-)amalgamation property. Then, we provide an equivalent syntactic characterization and show that such characterization covers most theories commonly employed in verification. Finally, we design a combined quantifier-free interpolation algorithm capable of handling both convex and nonconvex theories; this algorithm subsumes and extends most existing work on combined interpolation.

Abstract. In this paper we present a new decision procedure for the satisfiability of Linear Arithmetic Logic (LAL), i.e. boolean combinations of propositional variables and linear constraints over numerical variables. Our approach is based on the well known integration of a propositional SAT procedure with theory deciders, enhanced in the following ways.First, our procedure relies on an incremental solver for linear arithmetic, that is able to exploit the fact that it is repeatedly called to analyze sequences of increasingly large sets of constraints. Reasoning in the theory of LA interacts with the boolean top level by means of a stack-based interface, that enables the top level to add constraints, set points of backtracking, and backjump, without restarting the procedure from scratch at every call. Sets of inconsistent constraints are found and used to drive backjumping and learning at the boolean level, and theory atoms that are consequences of the current partial assignment are inferred.Second, the solver is layered: a satisfying assignment is constructed by reasoning at different levels of abstractions (logic of equality, real values, and integer solutions). Cheaper, more abstract solvers are called first, and unsatisfiability at higher levels is used to prune the search. In addition, theory reasoning is partitioned in different clusters, and tightly integrated with boolean reasoning.We demonstrate the effectiveness of our approach by means of a thorough experimental evaluation: our approach is competitive with and often superior to several state-of-the-art decision procedures. Motivations and GoalsMany practical domains require a degree of expressiveness beyond propositional logic. For instance, timed and hybrid systems have a discrete component as well as a dynamic evolution of real variables; proof obligations arising in software verification are often boolean combinations of constraints over integer variables; circuits described at Register Transfer Level, even though expressible via booleanization, might be easier to analyze at a higher level of abstraction (see e.g. [15]). Many of the verification problems arising in such domains can be naturally modeled as satisfiability in Linear Arithmetic Logic (LAL), i.e., the boolean combination of propositional variables and linear constraints over numerical variables. For its practical relevance, LAL has been devoted a lot of interest, and several decision procedures exist that are able to deal with it (e.g., SVC [17], ICS [24,19], CVCLITE [17,10], UCLID [36,33], HDPLL [30]).In this paper, we propose a new decision procedure for the satisfiability of LAL, both for the real-valued and integer-valued case. We start from a well known approach, previously applied in MATHSAT [26,4] and in several other systems [24,19,17,10, 35,3,21]: a propositional SAT procedure, modified to enumerate propositional assignments for the propositional abstraction of the problem, is integrated with dedicated theory deciders, used to check consistency of propositional assignments wit...

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